- Researchers train deep neural networks on databases that contained three-body problems and their solutions.
- Surprisingly, the network predicts accurate solutions at a fixed computational cost and up to 100 million times faster than the existing solver.
For over 3 centuries, mathematicians and physicists have puzzled over the three-body problem: a problem of calculating the motion of three bodies moving under no influence other than that of their mutual gravitation.
More specifically, if you take the initial positions and velocities of three-point masses and solve for their succeeding motions according to Newton’s laws of motion and universal gravitation, you will not find any general solution.
This is what a three-body problem is. Unlike two-body problems, there is no general closed-form solution, except for a small set of simple scenarios like identical planets moving in identical orbits.
Although the invention of powerful computers has enabled physicists to iteratively evaluate the positions of these point masses, it requires an extremely large number of computations resources. And even then, solutions remain vague.
To efficiently tackle this issue, researchers at the University of Edinburgh in Scotland have utilized an artificial intelligence (AI) model. Surprising, they were able to extract accurate solutions at a fixed computational cost and up to 100 million times faster than the existing solver.
Training And Validating Neural Network
The research team trained neural networks on a database of three-body problems. This database contained solutions computed by a novel solver.
To keep things simple, they started with simple problems that involved three bodies with equal mass and zero initial velocity. They selected arbitrary starting points and solved the three-body motion using a novel method named Brutus. This process was repeated ten thousand times.
They used 9,900 samples to train the neural network and 100 to validate it. To test this network, they then executed 5,000 entirely new scenarios and compared results to those computed by Brutus.
The network doesn’t actually calculate the future motion of three bodies, instead, it accurately predicts the future motion (using the knowledge gained in training phase). More specifically, it emulates the divergence between neighbor trajectories, which closely matches the Brutus simulations.
Simulation of a 3D body problem
In this study, the deep artificial neural network’s predicted solutions over a fixed time interval and met the energy conservation conditions with an error of 0.00001
This type of network can be used in situations when three-body problems become computationally unfeasible for Brutus. It could be part of a hybrid system where Brutus will perform all heavy calculations but when things go out of control, the network will step in until the situation becomes acceptable again.
For example, neural networks can be used to accurately simulate the motion of celestial objects inside globular star clusters and galactic nuclei, using less computational resources.
It is also possible to train neural networks on more complex problems, including 4 and 5-body problems, to decrease the computational burden up to a great extent.